D. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer Aided-Design, 10 (1978), pp. 356--360.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....all shape quantities to be optimized. Finally, such an explicit representation can serve as a good starting point for more expensive shape improving computations. 3.1. Generalized Subdivision. For the last few years, progress in modeling has been associated with generalized subdivision surfaces [CC78, DS78, PR97, Loo87, Kob00, WW01] as a natural averaging paradigm. The conceptual simplicity of subdivision surfaces has taken the entertainment industry by storm [DKT98] adding an additional representation to all standard graphics design packages. Yet, at present, no one seems to be ready to base a full car body design or the ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. ComputerAided Design, 10:356--360, September 1978.

....Biermann et al. 9, 10] also suggested the modification of the scheme to introduce sharp features. Stam [88] proposed an exact evaluation of the scheme without explicit subdivision process by utilizing eigenvector spaces of subdivision matrices. Doo Sabin Scheme In the same year, Doo and Sabin [30] also proposed new approximating subdivision surfaces on quadrilateral meshes. The limit of this process results in a bi quadratic B splines. Unlike the Catmull Clark scheme, they use the vertex split method to generate sub faces. Instead of introducing 4 sub faces for each quadrilateral, each ....

....and Warren [44] and Peters and Reif [70] proposed even simpler version of the Doo Sabin scheme, that is called Midedge subdivision which only uses 3 out of 4 vertices that described above. The regular rule can be reduced to 1 1 4 (p 1 p 2 ) 19) In addition to the scheme, Doo and Sabin [30] s work also has an important role in the subdivision analysis. They introduced the usage of subdivision matrix and Discrete Fourier Transform (DFT) as a main tool to analyze subdivision scheme at extraordinary vertices. Loop Scheme Figure 6: An example of the Loop subdivision surface. Image ....

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, Sept. 1978.

....that, throughout this paper the boundary of solid models is colored blue, and the cross section of solid models is colored yellow. velop a novel subdivision scheme for solid models that can interpolate all the control points scattered in 3D. 1.2. Subdivision Analysis Since Doo and Sabin s work [4], there has been much progress in analysis of subdivision schemes. Mostly, the analysis has been concentrated on extraordinary cases. Much literature [15, 17, 19] demonstrates proofs of continuity of various schemes, mainly based on spectral analysis techniques. Reif [15] discusses the necessary ....

.... Continuity over Irregular Topologies Unlike surface subdivision schemes whose irregular analysis involves only extraordinary vertices, we must take care of both extraordinary vertices and edges in solid schemes [1] Unfortunately, existing spectral analysis using Discrete Fourier Transform (DFT) [4, 19] cannot be directly adopted for solid schemes, as the technique is based on spectral behavior over a 2 dimensional domain. However, we can still employ eigenvalue and characteristic map analysis [15] numerically, at least for restricted cases, which are well understood techniques for surface ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, Sept. 1978. (d) (e) (f)

....claims with a number of simulations which exhibit characteristic effects of real world experiments. 1 Introduction Subdivision is an attractive method for free form geometric modeling. It elegantly addresses the classic challenge of building (piecewise) smooth surfaces of arbitrary topology [7, 10] and dovetails nicely with many modern algorithms which exploit multiresolution (for an overview see [32] In applications such as engineering design and animation it is also necessary to model the (dynamic or static) mechanical response of such surfaces subject to appropriate material models and ....

DOO, D., AND SABIN, M. Behaviour of Recursive Division Surfaces Near Extraordinary Points. Computer Aided Design 10 (1978), 356--360.

....This process can be automated by segmentation techniques, for example watershed segmentation, to get the areas in the mesh that need to be subdivided. We compare our methods for various triangular meshes and present our results. 1 Introduction When Catmull and Clark[i] and Doo and Sabin[2] published their papers little did they expect that subdivision would be used so extensively as it is being used today for the purposes of modeling and animation. It has been used to a large extent in movie production, commercial modelers such as MAYA 3.0, LIGHTWAVE 6.0 and game development ....

....in the field of computer graphics and computer aided geometric design(CAGD) mainly because it easily addresses the issues raised by multiresolution techniques to address the challenges raised for modeling complex geometry. The subdivision schemes introduced by Catmull and Clark[l] and Doo and Sabin[2] set the tone for other schemes to follow and schemes like Loop[6] Butterfly[3] and Modified Butterfly[14] Kobbelt [4] have become popular. These schemes are chiefly classified as either approximat ing, where the original vertices are not retained at newer levels of subdivision, or ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordi- nary points. Computer Aided Design, 10:356-360, 1978.

....Surfaces In this paper we deal exclusively with Catmull Clark subdivision surfaces. Nevertheless the ideas expressed here carry over to other subdivision surfaces as long as the derivative integrals over regular regions exist. This is, for example, the case for Doo Sabin subdivision surfaces [4], which in general are only C . Stam [18] showed that one can represent a Catmull Clark subdivision patch with a single irregular vertex of valencek (Figure 1) as an expansion in =2k 8functions S(u1,u 2) C i # i (u,u2 ) 4) The # i : are the eigen basis functions of the local ....

DOO, D., AND SABIN, M. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10, 6 (September 1978), 356--360.

....purpose. The first basic principle for subdivision was developed by Chaikin [26] in 1974: a smooth curve is generated through successive refinement of a polygonal curve defined with a set of control nodes, by inserting new nodes in between the old ones. Catmull and Clark [3] and Doo and Sabin [4] extended this principle to surfaces in 1978. A smooth surface is defined by infinite subdivision of an initial control mesh, subdivision still consisting in inserting new elements (i.e. nodes, faces and associated edges) Much effort has been made to ensure that subdivision processes converge ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10(6):356-- 360, 1978.

....ways. Either the polygonal domain, which is to be mapped into 3D, is subdivided in the parametric plane, or one uniform equation is used to represent the entire patch. In the former case triangular or rectangular elements are put together [2,6,12,20,23] or recursive subdivision methods are applied [5,8,24]. In the latter case either the known control point based methods are generalized or a weighted sum of 3D interpolants gives the surface equation [1,3,4,22,26] The method presented in this paper is a recursive subdivision scheme specially designed to consider arbitrary boundary conditions. ....

....and the topological information of the net E, in terms of edges and faces. A net is closed when each edge is shared by exactly two faces. Camull Clark s subdivision scheme is defined over closed nets of arbitrary topology, as an extension of the tensor product bi cubic B spline subdivision scheme [5,8]. Variants of the original scheme were analyzed by Ball and Storry [24] Our algorithm employs a variant of Catmull Clark s scheme due to Sabin [21] which generates limit surfaces that are C continuous everywhere except at a finite number of irregular points. In the neighborhood of those ....

D. Doo and M. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10:356--360, 1978.

....in the parametric plane, or one uniform equation is used as a com adilev math.tau.ac.il, http: www.math.tau.ac.il adilev bination of 3D constituents. In the former case triangular or rectangular elements are put together [2, 6, 12, 20, 23] or recursive subdivision methods are applied [5, 8, 24]. In the latter case either the known control point based methods are generalized or a weighted sum of 3D interpolants gives the surface equation [1, 3, 4, 22, 26] This paper presents a subdivision schemes specially designed for the task of filling n sided holes, which belongs to the former ....

....topological information of the net E, in terms of edges and faces. A net is closed when each edge is shared by exactly two faces. Camull Clark s subdivision scheme is defined over closed nets of arbitrary topology, as an extension of the tensor product bi cubic B spline subdivision scheme (see [5, 8]) Variants of the original scheme were analyzed by Ball and Storry [24] Our algorithm employs a variant of Catmull Clark s scheme due to Sabin [21] which generates limit surfaces that are G continuous everywhere except at a finite number of irregular points. In the neighborhood of those ....

D. Doo and M. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10:356--360, 1978.

....implementation of this idea in several cases. For each example, we constructed special schemes that operate near the boundary and we checked their smoothness using the theory developed in [3] 2 Interpolating smooth boundary curves with Catmull Clark s scheme Catmull Clark s scheme is defined in [1] over meshes with no boundaries, as follows: Every face of the mesh is divided into 4 sided faces , as shown in figure 1. There are three types of vertices in the resulting mesh (see figure 2) a) A vertex that corresponds to an old face is calculated by averaging the control points ....

D. Doo and M. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10:356--360, 1978.

....of the analysis techniques on R n , e.g. the Fourier transform. In Section 2.4 we introduced the key tools for analysis of subdivision surfaces, the subdivision map (or subdivision matrix) and the characteristic map. Since subdivision surfaces were introduced by Catmull, Clark, Doo and Sabin in [4, 7] the analysis of these surfaces has been based on the subdivision matrix. The characteristic map, derived from the eigenvectors of the subdivision matrix, was introduced by Reif in [16] He proved that if the characteristic maps are regular and injective, then subdivision surfaces are C 1 for ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10(6):356--360, 1978.

....our findings and describe several avenues of future research. 2 Subdivision Surfaces In 1974 Chaikin [4] introduced the idea of generating a curve from a polygon by successively refining the polygon with the addition of new vertices and edges. In 1978, Catmull and Clark [2] and Doo and Sabin [6] generalized the idea to surfaces. In these schemes, an initial control mesh is refined by adding new vertices, faces and edges at each subdivision step. In the limit as the number of subdivision steps goes to infinity, the control mesh converges to a surface. With careful choice of the rules by ....

....M 0 such that the subdivision surface it defines interpolates some or all of the vertices of b I. It is also possible to constrain the surface to have a specified normal at each interpolation point. Nasri [12] generates interpolating surfaces using the biquadratic formulation of Doo and Sabin [6]. Like biquadratic B splines, Doo Sabin surfaces interpolate the centroid of each face in the control mesh. Thus a linear constraint on the control vertices can be generated for each interpolation point and the resultant system solved for the desired control mesh 1 . It appears that Nasri had no ....

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10(6):356--360, 1978.

....pre condition for constructing wavelet spaces can be generated by linear subdivision (see [14] for details) The limit function may be calculated directly. It is not required to perform any iterations. The first and most popular subdivision schemes for surfaces were introduced by Doo Sabin [4] and Catmull Clark [1] Their methods are both based on quadrilateral meshes and either generalize biquadratic or bicubic tensor product B splines. The simplest scheme for triangle meshes, introduced by Loop [12] splits each triangle into four and converges to quartic triangular B Splines. All ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, September 1978.

....pre condition for constructing wavelet spaces can be generated by linear subdivision (see [20] for details) The limit function may be calcu lated directly. It is not required to perform any iterations. The first and most popular subdivision schemes for surfaces were introduced by Doo Sabin [8] and Catmull Clark [2] Their methods are both based on quadrilateral meshes and either generalize biquadratic or bicubic tensor product Bsplines. The simplest scheme for triangle meshes, introduced by Loop [17] splits each triangle into four and converges to quartic triangular B splines. Figure ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, Sept. 1978.

....smoothing steps (# #0#5) Note the shrinkage effect. D) Five non shrinking smoothing steps (k PB #0#1and # #0#6307) of this paper. B) C) and (D) are the surfaces obtained after two levels of refinement and smoothing. Surfaces are flat shaded to enhance the faceting effect. subdivision schemes [8, 4, 12] are rigid, in the sense that they have no free parameters that influence the behavior of the algorithm as it progresses trough the subdivision process. By using our fairing algorithm in conjunction with subdivision steps, we achieve more flexibility in the design process. In this way our fairing ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10:356--360, 1978.

....3D File Formats, Geometry Compression, Algorithms, Graphics. 1 INTRODUCTION Subdivision surfaces are becoming popular multi resolution representations in modeling and animation [22, 21] The most popular uniform recursive subdivision schemes are due to Loop [11] Catmull Clark [2] and Doo Sabin [3]. For example Figure 1 B shows the result of applying Loop s triangle quadrisection scheme [11] to the triangular mesh shown in Figure 1 A. Since the most popular interchange le formats, such as VRML [19] do not preserve the subdivision structure, a problem exists if the model is saved using one ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10:356--360, 1978.

....for an effective multiresolution decomposition of unstructured meshes is the flexible representation of detail information. We discuss several approaches. 1 Introduction Subdivision techniques provide very efficient and flexible algorithms for the generation of free form surface geometry [2, 5, 6, 18, 25, 39]. Starting with an arbitrary control mesh 0 we can apply the subdivision rules to compute finer and finer meshes m with control vertices p m i becoming more and more dense until the desired approximation tolerance required for a given application is reached. The result is a smooth surface ....

DOO, D., AND SABIN, M. Behaviour of recursive division surfaces near extraordinary points. CAD (1978).

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356--360, 1978.

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DOO D., SABIN M.: Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design 10, 6 (1978), 177--181.

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D. Doo and M. A. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10(6):356--360, 1978.

....of the points of S 1 with an image of S 1 scaled down by a factor of n. Continuing recursively we define the set P P i Sm 1 , i = 0, 1, k (4) original point in S 1 in S 2 in S 3 original point in S 1 in S 2 , in S 3 Figure 3: The first four point sets for (a) Doo Sabin [5] and (b) Loop [9] schemes. The black dot in the center represents the original vertex under consideration. The pale grey dots represent other vertices of the original grid, thereby showing the relationship between the original grid and the point sets. Note that, in the Loop scheme, S 1 contains ....

....obtained from Sm 1 by substituting each of its points with an image of S 1 scaled down by a factor of n . Sm can also be written P i 1 P i 2 . i 1 , i 2 , i m = 0, 1, k (5) Figure 3 illustrates the point sets S 1 , S 2 , and S 3 for the Doo Sabin quadrilateral [5] and Loop triangular [9] schemes respectively. To find the total support we have to define a limit for the sequence of sets (1) and then take the closure, that is S = lim m## Sm (6) Figure 4 shows two interesting cases with S 4 offering a good approximation to S. Notice that the topological ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356--360, 1978.

....Also, it is obvious from the above tables that alternative, more economic, codifications are possible. For example, the added fractions in the coordinates of the point P 1 in the cases TD,QD,QM can be omitted as they convey no information. With this abbreviated notation the familiar Doo Sabin [5] scheme follows the refinement pattern QD(2,0) the Catmull Clark [2] is QP(2,0) the Loop [11] and the Butterfly [7] are TP(2,0) and the # 3 scheme is TP(1,1) Some further shortening of the noatation may be achieved with the unification of P and M cases in a more more compact but less ....

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356--360, 1978.

....necessary conditions for the corresponding properties: A B C D E F a b c d e f Figure 3: Configuration around mid point. kink i.e. not C 1 # 2 unbounded curvature mildly diverging curvature curvature bounded 1 (4) This analysis was first demonstrated in [2], in which the terminology is explained more fully. For the purposes of this paper we are interested in showing that , i.e. that the curvature of the limit function is bounded, which is a necessary condition if the limit function is to have C continuity. We shall perform the analysis ....

D. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer-Aided Design 10 (1978), 356--360. This is the 6-point scheme presented in [9]

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D. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer Aided-Design, 10 (1978), pp. 356--360.

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Doo, D., and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer-Aided Design 10 (1978) 356--360; reprinted in Seminal Graphics, ACM, New York (1998) 177--181.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, 1978.

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Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10 (1978) 356--360

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356--360, September 1978.

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DOO, D., AND SABIN, M. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design 10 (1978), 356--360.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10(6):356--360, 1978.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Comput. Aided Design, 10:356--360, 1978.

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D. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer-Aided Design, Vol 10, No. 6, 1978, pp. 356--360.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. ComputerAided Design, 10:356--360, September 1978.

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D. Doo and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer-Aided Design, Vol. 10, No. 6, Nov. 1978, pp. 356--360.

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DOO,D.AND SABIN, M. 1978. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10, 6, 356--360.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356---360, September 1978.

No context found.

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, September 1978.

No context found.

Doo, D., and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer-Aided Design 10 (1978) 356--360; reprinted in Seminal Graphics, ACM, New York (1998) 177--181.

No context found.

DOO, D., AND SABIN, M. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10, 6 (September 1978), 356--360.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10(6):356--360, September 1978.

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D Doo and M Sabin. Behaviour Of Recursive Division Surfaces Near Extraordinary Points. Computer-Aided Design, 10:356--360, 1978.

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D. Doo and M. Sabin. Behaviour of Recursive Division Surfaces Near Extraordinary Points. Computer Added Design, 10(6):356--360, 1978.

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Doo, D., Sabin, M.: Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design 10 (1978) 356--360

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D. Doo and M. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10:356-360, 1978.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer Aided Design, 10(6):356--360, 1978.

No context found.

D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Computer-Aided Design, 10:356---360, September 1978.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. Comput. Aided Design, 10:356--360, 1978.

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Doo, D. W. H. and M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer Aided Design 10 (1978), 356--360.

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D. Doo and M. Sabin. Behaviour of recursive division surfaces near extraordinary points. ComputerAided Design, 10:356--360, September 1978.

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Doo, D., and Sabin, M. Behaviour of Recursive Division Surfaces Near ExtraordinaryPoints. Comput. AidedDesTA 10, 6(1978), 356--360.

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